On loop spaces of configuration spaces

This article gives an analysis of topological and homological properties for loop spaces of configuration spaces. The main topological results are given by certain choices of product decompositions of these spaces, as well as ``twistings" between the factors. The main homological results are given in terms of extensions of the ``infinitesimal braid relations" or ``universal Yang-Baxter Lie relations".

     

On steiner spaces

A Steiner space is a Steiner triple system that is not generated by a triangle. We give new constructions of Steiner spaces and solve the existence problem of Steiner spaces of order v for all but 4 values of v.     

On c-realcompact spaces

Abstract

The c-realcompact spaces are fully studied and most of the important and well-known properties of realcompact spaces are extended to these spaces. For a zero-dimensional space X, the space υ₀X, which is the counterpart of υX, the Hewitt realcompactification of X, is introduced and studied. It is shown that υ₀X, which is the smallest c-realcompact space between X and β₀X, plays the same role (with respect to C_c(X)) as υX does in the context of C(X). It is proved for strongly zero-dimensional spaces, c-realcompact spaces, realcompact spaces and N-compact spaces coincide. In particular, if X is a strongly zero-dimensional space, then υX = υ₀X. It is obsesrved that a zero-dimensional space X is pseudocompact if and only if C_c(X) = C*_c(X), or equivalently if and only if υ₀X = β₀ X. In particular, a zero-dimensional pseudocompact space is compact if and only if it is c-realcompact. It is shown that Lindelöf spaces, subspaces of the one-point compactification (resp., Lindelöffication) of a discrete space with a nonmeasurable cardinal, are c-realcompact space. If X is a pseudocompact space, it is observed that C(X) = C_c(X) if and only if βX is scattered. Finally, the simplest possible proof (with reasoning) among the known proofs, of the well-known fact that discrete spaces of cardinality less than or equal to that of the continuum are realcompact, is given.     

On gauss spaces

Summary Theorem 1 gives a characterization of the Gauss spaces. As a consequence of this theorem we construct some examples and counter-examples of such spaces. Finally, we define and sludy some properties of the canonical standard family of coverings, Gauss subspace, Guss product space, etc. Entrata in Redazione il 6 luglio 1968.     

On spaces of functions between classifying spaces

In this paper the results of Dwyer and Zabrodsky [DZ] are extended by showing that ifL is a compact Lie group andG is either ap-group or a torus, then every mapf:BG →BL is homotopic to one induced by a homomorphismφ :G →L, and two such induced maps are homotopic if and only if the corresponding homomophisms are conjugate. Several other results related to maps between classifying spaces, completions, and fibrations are also deduced.     

On certain co-H spaces related to Moore spaces

We show that certain co- spaces, constructed by Anick and Gray, carry a homotopy co-associative and co-commutative co- structure.

     

On banach spaces whose duals areL 1 spaces

A structure theorem for Banach spaces whose duals areL ₁ spaces, is proved. The research of the second named author has been sponsored by the Air Force Office of Scientific Research under Grant AF EOAR 66-18 through the European Office of Aerospace Research (OAR) United States Air Force.     

On fuzzy topological spaces

In this note the concept of almost fuzzy continuous mapping from a fuzzy topological space into another is introduced and discussed.     

On separable Lindenstrauss spaces

Isometric embeddings from l_∞ⁿin l_∞^{n + 1} can be described by a_i,n, i ? n, with $\text{∑}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{n}}\text{¦ a}{}_{\mathrm{i,n}}\text{¦ ? 1}$, such that e_i,n = e_{i,n + 1} + a_i,ne_{n + 1,n + 1}; i = 1,…, n; holds, where e_i,nand e_{i,n + 1} are the elements of the canonical unit vector bases of l_∞ⁿand l_∞^{n + 1}, respectively (negative signs may occur). We study the connections between a triangular substochastic matrix A, whose nth column consists of the elements a_i,n, i = 1,…, n, and the Banach space a_i,n, E_n ? E_{n + 1}, E_n ? l_∞ⁿ, where A determines the embeddings of the E_n. The class of these Banach spaces is the class of all separable Lindenstrauss spaces. Sufficient and necessary conditions are stated for a matrix A to represent c₀and c. Furthermore, we characterize the class of all extreme triangular substochastic matrices which represents C(K), where K is the Cantor set. We investigate how the special biface structure of the dual unit ball of X is reflected in the elements of a matrix A representing the separable Lindenstrauss space X. This is applicable to Gurarij spaces; we give a new proof for the maximality property of Gurarij spaces and show that they are isomorphic to A(S) where S is a Choquet simplex with dense extreme points.     

On probabilistic commutative spaces

A new characterization is given for the probabilistic commutative Riemannian spaces studied originally (in the compact case) by Roberts and Ursell. Moreover, some earlier basic results are generalized to the non-compact case.     

On spaces of nearexistence

The notion of subspace of nearexistence is introduced. In particular, it is proved that ifQ is a countable compact set, then any subspaceL C(Q), dimL=codimL=+, can be approximated by subspaces of nearexistence.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 278–287, August, 1996.This work was supported by the Fundamental Problems of Mathematics and Mechanics Foundation under grant No. 1.1.64.